## Type of Work

Article (1196) Book (0) Theses (22) Multimedia (2)

## Peer Review

Peer-reviewed only (1193)

## Supplemental Material

Video (0) Audio (0) Images (1) Zip (1) Other files (5)

## Publication Year

## Campus

UC Berkeley (888) UC Davis (53) UC Irvine (947) UCLA (50) UC Merced (4) UC Riverside (11) UC San Diego (61) UCSF (61) UC Santa Barbara (24) UC Santa Cruz (887) UC Office of the President (43) Lawrence Berkeley National Laboratory (977) UC Agriculture & Natural Resources (0)

## Department

Research Grants Program Office (RGPO) (40) University of California Research Initiatives (UCRI) (1) UC Lab Fees Research Program (LFRP); a funding opportunity through UC Research Initiatives (UCRI) (1)

UCLA Graduate School of Education and Information Studies (6) Bourns College of Engineering (4) Center for Environmental Research and Technology (4)

## Journal

Western Journal of Emergency Medicine: Integrating Emergency Care with Population Health (4) Berkeley Scientific Journal (3) Dermatology Online Journal (2) International Conference on GIScience Short Paper Proceedings (1)

## Discipline

Engineering (23) Medicine and Health Sciences (11) Life Sciences (9) Social and Behavioral Sciences (5) Physical Sciences and Mathematics (3) Architecture (1) Arts and Humanities (1) Business (1) Education (1) Law (1)

## Reuse License

BY - Attribution required (717) BY-NC-ND - Attribution; NonCommercial use; No derivatives (4) BY-NC - Attribution; NonCommercial use only (2) BY-NC-SA - Attribution; NonCommercial use; Derivatives use same license (2) BY-ND - Attribution; No derivatives (1)

## Scholarly Works (1222 results)

The big bang theory is a model of the universe which makes the striking prediction that the universe began a finite amount of time in the past at the so called "Big Bang singularity." We explore the physical and mathematical justification of this surprising result. After laying down the framework of the universe as a spacetime manifold, we combine physical observations with global symmetrical assumptions to deduce the FRW cosmological models which predict a big bang singularity. Next we prove a couple theorems due to Stephen Hawking which show that the big bang singularity exists even if one removes the global symmetrical assumptions. Lastly, we investigate the conditions one needs to impose on a spacetime if one wishes to avoid a singularity. The ideas and concepts used here to study spacetimes are similar to those used to study Riemannian manifolds, therefore we compare and contrast the two geometries throughout.

Cryptographic assumptions and security goals are fundamentally distributional. As a result, statistical techniques are ubiquitous in cryptographic constructions and proofs. In this thesis, we build upon existing techniques and seek to improve both theoretical and practical constructions in three fundamental primitives in cryptography: blockciphers, hash functions, and encryption schemes. First, we present a tighter hybrid argument via collision probability that is more general than previously known, allowing applications to blockciphers. We then use our result to improve the bound of the Swap-or-Not cipher. We also develop a new blockcipher composition theorem that is both class and security amplifying. Second, we prove a variant of Leftover Hash Lemma for joint leakage, inspired by the Universal Computational Extractor (UCE) assumption. We then apply this technique to construct various standard-model UCE- secure hash functions. Third, we survey existing “lossy primitives” in cryptography, in particular Lossy Trapdoor Functions (LTDF) and Lossy Encryptions (LE); we pro- pose a generalized primitive called Lossy Deterministic Encryption (LDE). We show that LDE is equivalent to LTDFs. This is in contrast with the block-box separation of trapdoor functions and public-key encryption schemes in the computational case. One common theme in our methods is the focus on statistical techniques. Another theme is that the results obtained are in contrast with their computational counterparts—the corresponding computational results are implausible or are know to be false.

In this thesis, we study linear stability of Einstein metrics and develop the theory of Perelman's $\lambda$-functional on compact manifolds with isolated conical singularities. The thesis consists of two parts. In the first part, inspired by works in \cite{DWW05}, \cite{GHP03}, and \cite{Wan91}, by using a Bochner type argument, we prove that complete Riemannian manifolds with non-zero imaginary Killing spinors are stable, and provide a stability condition for Riemannian manifolds with non-zero real Killing spinors in terms of a twisted Dirac operator. Regular Sasaki-Einstein manifolds are essentially principal circle bundles over K\"{a}hler-Einstein manifolds. We prove that if the base space of a regular Sasaki-Einstein manifold is a product of at least two K\"{a}hler-Einstein manifolds, then the regular Sasaki-Einstein manifold is unstable. More generally, we show that Einstein metrics on principal torus bundles constructed in \cite{WZ90} are unstable, if the base spaces are products of at least two K\"{a}hler-Einstein manifolds.

In the second part, we prove that the spectrum of $-4\Delta+R$ consists of discrete eigenvalues with finite multiplicities on a compact Riemannian manifold of dimension $n$ with a single conical singularity, if the scalar curvature of cross section of conical neighborhood is greater than $n-2$. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectrum properties, we extend the theory of Perelman's $\lambda$-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.